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PERIODIC MOTION                           303
7.   A particle which is constrained to move in a straight line is attracted by another particle fixed at a point outside the line.   Show that }he motion of the particle is simple harmonic when the force varies as the listance between the particles. •
8.   A particle of mass m describes a motion defined by the equation
x = a sin (ut + 5).
Find the average value of the following quantities, with respect to the ;ime, for an interval of half the period:
(a)  displacement;                        (e) momentum;
(b)  velocity;                               (f) kinetic energy;
(c)  acceleration;                         (g) potential energy.
(d)  force;
9.   In problem 8 take the averages with respect to position.
10.    In problem 8 suppose the motion to be given by
x — a cos w (t + to).
11.    In problem 10 take the averages with respect to position.
12.    In problem 8 suppose the motion to be given by the following equations:
I.   x ~ a sin2 (co£ + 5). II.   x = a cos2 (ut + 5). III.   x — a sin ut cos (ut + 5).
238.   Composition of Two Parallel Simple Harmonic Motions af Equal Period.   Analytical Method. — Suppose
Xi = ai sin ((at + 5i),                             (1)
and                        z2 = a2 sin (ut + &),                          (2)
to define the motions which a particle' would have if acted upon, separately, by two simple harmonic forces. Then the motion which will result when the forces act simultaneously is obtained by adding equations (1) and (2). Thus
x = Xi + x2 = di sin («< + Si) + a2 sin (ut + 52).